Jacobian conjecture

The Jacobian conjecture asserts that every polynomial map over a field of characteristic zero with a nonzero constant Jacobian determinant has a polynomial inverse. It remains open in dimensions greater than one.

Polynomial map: A map between affine spaces whose coordinate functions are polynomials. The conjecture asks when such a map has a polynomial inverse.
Formal power series: Infinite algebraic sums whose coefficients are manipulated without requiring numerical convergence. At each point, an invertible linear term yields a formal inverse expansion.
Ott-Heinrich Keller: A German mathematician who posed the polynomial invertibility problem now known as the Jacobian conjecture. His 1939 paper introduced the question for polynomial maps with constant nonzero Jacobian.
Inverse function theorem: A theorem guaranteeing a local differentiable inverse where the derivative is invertible. It proves a local analogue, while the Jacobian conjecture demands a global polynomial inverse.
Jacobian matrix: The matrix of first partial derivatives of a differentiable map. Its determinant supplies the conjecture’s local invertibility condition.
Formal inverse function theorem: A theorem giving a formal power-series inverse when a map’s linear part is invertible. It explains why the constant Jacobian condition gives formal inverses locally.
Keller map: A polynomial map whose Jacobian determinant is a nonzero constant. Keller’s terminology names precisely the maps tested by the conjecture.
Birational map: A rational map that has a rational inverse on dense open subsets. Rational invertibility is weaker than the polynomial inverse required by the conjecture.
Jacobian determinant: The determinant of a map’s Jacobian matrix, encoding its local volume scaling and orientation. The conjecture assumes this polynomial is a nonzero constant.
Étale morphism: A morphism that is flat and unramified, serving as an algebraic analogue of a local isomorphism. Polynomial maps with invertible Jacobian are étale, connecting the conjecture to algebraic geometry.

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